This paper analyses the implementation of the generalized
finite differences method for the HJB equation of
stochastic control, introduced by two of the authors in
[Bonnans and Zidani,
SIAM J. Numer. Anal.41 (2003) 1008–1021]. The computation of coefficients needs to
solve at each point of the grid (and for each control)
a linear programming problem.
We show here that, for two dimensional problems, this
linear programming problem can be solved in O(pmax)
operations, where pmax is the size of the stencil.
The method is based on a walk on the Stern-Brocot tree,
and on the related filling of the set of
positive semidefinite matrices of size two.

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